Like icarus devined, A is the area of the cusps (perpendicular to the field). For the line cusps, it is their total length times twice the thickness of the sheath. Using conservation of flux in the sheath, adding the area of the point cusps should double this.

This may be the most interesting problem I have come across since I have been hanging out here. I thought Bohm was going to get my back, but it turns out he was relying heavily on Boltzmann, who is persona non grata here. I thought I could whip up a substitute theory on the basis of monoenergetic ions and electrons (plus a few extras later on), but it turns out that there is no solution to the sheath problem under those conditions. (Nota Bene, no solution at all, not simply one I don't like.)

Actually the Bohm condition arises in a similar way. One asks what the sheath structure should be, given the ion velocity, and one discovers that there is no solution unless V_i >= c_s. Physically, you might set up a situation with V_i < c_s, thinking nature will surely think of something to do. What occurs to nature is to let the sheath eat back into the plasma to provide a potential hill to accelerate the ions to the speed they need to have when entering the real sheath. This hill, known as the "pre-sheath", is closely related to the particle sources, which we were trying to ignore.

It could be that the sheath in this case will have some long range effects that end up broadening the electron energies so they are no longer so pathologically narrow.

I could retreat to my conviction that the electrons will be Maxwellian, or I could be satisfied that nobody else has a good theory either. I have a bit more ambition than that, so I'll keep trying, but it may take a while.

It is possible that since the Magrid is positively biased the plasma sheath (outer layer) is made up of negatively charged particles, i.e. electrons. It may be nothing like the classical positive ion Debye sheath surrounding a solid-surface negatively charged electrode that the Bohm criterion derivation is based upon.

The vacuum and magnetic field between the Magrid and plasma may just serve to provide a standoff distance but the plasma layer derivation needs to begin from the point of a plasma surrounding the solid-surface of a positively charged electrode. We could call it a "Carlson sheath" if you can pull it off.

Here's where I'm coming from. Obviously, I know what's in the data and I know what it is consistent with. We've known that for several months. However, just because a piece of data is consistent with one model doesn't mean that it's inconsistent with another model. When you compare data and theory, the best you can say is "it's consistent with...." .

I know that the confinement in the WB-7 is much, much better than ballistic. I also believe that the dominant energy flowthrough is in the electrons, not the ions. Glen Wurden (whom you probably know from LANL) has taken some fast framing pictures of the plasma, and the hot spots are on the coils, not the walls. While this isn't a proof, it's a pretty strong indicator.

That's why I'm interested in how big A is and how it scales. It's conceivable that even if ion losses aren't dominating right now, they might bite us in the behind down the road. It would be useful to know what the knobs are and how to control them.

"Electron sheaths are electric fields that form near plasma boundaries to preserve quasineutrality in the bulk plasma by reducing the positive ion current lost to a boundary. Previous conjectures hypothesize that electron sheath formation near a positive anode is a solution to the current balance when the ratio of the anode area, Aa, to the chamber wall area, Aw, is less than the square root of the ion to electron mass ratio, Aa/A w<(me/mi)1/2. However, we show that the general nature of an electron sheath exhibits a potential dip which allows the electron sheath solution for a larger anode than is predicted by this relationship. Potential dips have been observed in electron sheaths, but were attributed to an ion pumping mechanism to an insulator present near the probe. We show that the dip is necessary even when no insulator is present and that the ion pumping needed for a steady-state potential well to exist is provided naturally by the plasma potential adjusting to decrease from the well bottom to the chamber wall. Data were taken for low-pressure argon plasma generated by hot filaments and confined in a multidipole chamber. Bulk plasma density, electron temperature and potential were measured with a planar Langmuir probe. Plasma potential profiles in the sheath regions were measured with an emissive probe operated in the limit of zero emission."

Art Carlson wrote:[*] The main ball and the cusp plasma will both be quasineutral.

I believe a major deviation from the Buzzard model, where the sheath and cusps deviate greatly from quasi-neutrality, the ions not having the energy to reach those regions. Any data on hand or from planned experiments to resolve this matter?

Here's where I'm coming from. Obviously, I know what's in the data and I know what it is consistent with. We've known that for several months. However, just because a piece of data is consistent with one model doesn't mean that it's inconsistent with another model. When you compare data and theory, the best you can say is "it's consistent with...." .

I always liked "it's not necessarily inconsistent with ....". The fact that you are the only one who's seen the data makes it difficult to carry on a discussion on the experimental side. Is there any chance something will be published soon? Is there any chance you can send your confidential report to me, if I promise to zip my lips?

rnebel wrote:I know that the confinement in the WB-7 is much, much better than ballistic. I also believe that the dominant energy flowthrough is in the electrons, not the ions. Glen Wurden (whom you probably know from LANL) has taken some fast framing pictures of the plasma, and the hot spots are on the coils, not the walls. While this isn't a proof, it's a pretty strong indicator.

By "ballistic", do you mean a transit time, or are you taking the cusp/surface ratio into account somehow? And you believe that "energy flowthrough in the electrons" is landing on the coils, rather than the walls? Give my regards to Glen when you see him again. He spent several months in Garching 20 years ago. I know the date because Fleischmann and Pons held their press conference and he started doing electrolysis and calorimetry.

rnebel wrote:That's why I'm interested in how big A is and how it scales. It's conceivable that even if ion losses aren't dominating right now, they might bite us in the behind down the road. It would be useful to know what the knobs are and how to control them.

My best guess at the area of a point cusp is

A_point = 4*R*rho_e

There are 6 point cusps and the line cusps will have a similar loss area, so the total area of the cusps should be

A_tot = 50*R*rho_e

R is the radius of the plasma, which I guess will be something like half the side of the cube with the coils. I'm pretty confident that the density and temperature/energy in the cusps will be not too much lower than those in the plasma ball, but those need to be measured. Estimating them from the presumed beta=1 condition is a bit too indirect.

If the beta=1 condition represents the radius of the wiffleball (I'm assuming that's what you are using), then the plasma pressure at the boundary is always balanced by mag. pressure of the confinement field, independent of wiffleball radius.

If the beta=1 condition represents the radius of the wiffleball (I'm assuming that's what you are using), then the plasma pressure at the boundary is always balanced by mag. pressure of the confinement field, independent of wiffleball radius.

Shouldn't it be related to the square of electron-gyroradius?

A_point ~ pi*(rho_e^2)

This is what I learned from Haines. The sheath is spread over the whole surface of the ball. As it converges on the point cusp, the flux tubes gather together and the sheath gets thicker. I got 4*R*rho_e by calculating the flux where the sheath is thinnest. There it is perhaps as thin as rho_e, but it cuts through a circle with a radius related to R.